To construct a truth table for the given statement $\left(\sim p \lor \sim r \right) \leftrightarrow \left(\sim r \lor q \right)$, we will create a table with all possible combinations of truth values for $p$, $q$, and $r$. Then, we will compute the values of the sub-expressions step by step, which includes negating variables and evaluating the logical disjunction (OR, $\lor$) and biconditional ($\leftrightarrow$) operations.

Key Symbols:

• $\sim p$: Negation of $p$
• $\lor$: Logical OR
• $\leftrightarrow$: Biconditional ("if and only if")

Step-by-step process:

1. Column 1-3: Assign all possible truth values for $p$, $q$, and $r$.
2. Column 4: Compute $\sim p$ and $\sim r$, then find the value of $\sim p \lor \sim r$.
3. Column 5: Compute $\sim r$, then find the value of $\sim r \lor q$.
4. Column 6: Evaluate the biconditional $\left(\sim p \lor \sim r\right) \leftrightarrow \left(\sim r \lor q\right)$.

Let's build the truth table step-by-step.

$p$	$q$	$r$	$\sim p \lor \sim r$	$\sim r \lor q$	$\left(\sim p \lor \sim r\right) \leftrightarrow \left(\sim r \lor q\right)$
T	T	T	F	T	F
T	T	F	T	T	T
T	F	T	F	F	T
T	F	F	T	F	F
F	T	T	T	T	T
F	T	F	T	T	T
F	F	T	T	F	F
F	F	F	T	F	F

Explanation:

• $p$, $q$, and $r$: The first three columns list all possible combinations of truth values for $p$, $q$, and $r$ (T = true, F = false).
• $\sim p \lor \sim r$: This column shows the result of the disjunction (OR) between the negation of $p$ and the negation of $r$.
• $\sim r \lor q$: This column shows the result of the disjunction (OR) between the negation of $r$ and $q$.
• $\left(\sim p \lor \sim r\right) \leftrightarrow \left(\sim r \lor q\right)$: The biconditional checks if the two sub-expressions have the same truth value. If they do, the result is T (True); otherwise, it is F (False).

Let me know if you'd like more details or explanations!

5 Related Questions:

1. What does the symbol $\leftrightarrow$ mean in logical expressions?
2. How do you compute the negation ($\sim$) of a truth value?
3. What is the difference between logical OR ($\lor$) and logical AND ($\land$)?
4. How can you verify if two logical expressions are logically equivalent?
5. How does the truth table change if we modify one of the sub-expressions?

Tip:

Always break down complex logical expressions into smaller components when constructing truth tables!